The chatter in Democratic circles has been dominated by the news that Mr. Paterson discussed the Senate seat with Ms. Kennedy last week, holding open the possibility that she might be chosen to occupy the seat once held by her uncle Robert F. Kennedy, and serve alongside another uncle, Senator Edward M. Kennedy.

Mr. Paterson, for his part, has spent much of the past few weeks fielding questions about whom he will choose to replace Mrs. Clinton, who was picked to be secretary of state by President-elect Barack Obama. The governor made sport of the speculation during a speech in Washington on Saturday night.

But he said that though he and Ms. Kennedy had discussed the Senate seat, the ball was essentially in her court.

“I have enough qualified candidates that I don’t think I have to draft anybody,” he said in an interview on Sunday night. “Part of the consideration is how willing people are to serve, so I would figure if they were willing, they would make the request. I haven’t really taken anyone in hand and asked them.”

“Her questions were more informational,” he said, referring to their conversation last week. “She said, ‘I’ll talk to you again to let you know if I’m interested,’ and I thought that was fine.”

Mr. Obama was asked on Sunday about Ms. Kennedy on NBC’s “Meet the Press.”

“Caroline Kennedy has become one of my dearest friends, and is just a wonderful American, a wonderful person,” Mr. Obama said, “but the last thing I want to do is get involved in New York politics.”

Mr. Paterson said he would be meeting with Mrs. Clinton, Senator Charles E. Schumer and other officials this week to discuss the vacancy. He said he had not spoken to Senator Kennedy.

The governor has been able to make sport of the rampant speculation over who would replace Mrs. Clinton. He was one of the two main speakers, along with the Mississippi governor, Haley Barbour, on Saturday at the winter dinner of the Gridiron Club, a press association, where politicians often try their hand at humor.

He noted the cavalcade of luminaries bending his ear.

“They are blowing up my cellphone,” he said, according to a transcript, and then began listening to “messages” from New York’s top political figures that were recorded on his cellphone and played over the audio system.

Photo

Caroline Kennedy is said to be interested in Hillary Rodham Clintons seat in the Senate.Credit
Kristen Mullen/The Citizens’ Voice, via Associated Press

First, there was the voice of Mr. Schumer.

“Hi, Dave,” he said. “It’s Chuck Schumer. I thought I would let you know my view on the Senate race. I have one criteria above all. I would like a devout evangelical. That way, there will be no competition for Sunday press conferences.”

Mr. Schumer was mocking his penchant for holding news conferences on Sunday to get media coverage on what is generally a quiet news day.

“I know you’ll make a wise and careful choice, but Chuck asked me to make a suggestion,” she said. “There are so many qualified candidates. And Chuck knows that naming a replacement will be very difficult. So he figures, Why bother? Why does New York need two senators, anyhow?”

She went on at some length to say she knew there was “just one candidate who will bring the perfect combination of intelligence, integrity and passion to this job, and that person is...”

At that point, the voice mail beeped and the future secretary of state — pending Senate confirmation — was cut off.

Finally, the governor left himself a message.

“Hello, Governor,” he said. “It’s David Paterson here. I am calling to remind you that things would be a lot easier if you appoint yourself to the Senate seat.

“Leave that budget mess in Albany to some other sucker,” he said, adding, “I can’t believe you’re not thinking about this. I know I am.”

The governor, it should be noted, once did have his sights set on the seat — when he was lieutenant governor and Mrs. Clinton was the front-runner in the presidential race.

Now that he is governor, he has ruled out appointing himself.

His speech got high marks.

“Several members commented that he was one of the better speakers we’ve had,” said Carl Leubsdorf, the Washington bureau chief of The Dallas Morning News and president of the Gridiron. “He was very funny.”

Signing off, Mr. Paterson told the audience: “I ask all of you to turn your plate over. There may be a gold seal on the bottom of your plate. If you have that seal, you will be the next senator of the State of New York.”

A version of this article appears in print on , on Page A25 of the New York edition with the headline: Amid Talk of Kennedy, Paterson Turns to Humor. Order Reprints|Today's Paper|Subscribe